The big island is the location for the Ironman, and there are lots of cyclists in and around Kona. I rode most of the route before my race and I recall the roads being pretty good and no problems from the motorists.
I haven't ridden on any other islands, but I do recall the speed limits being relatively slow and thinking that in SOME places it would be fun to ride. Others, however, seemed to be quite congested. I would definitely look for a resource dedicated to cycling in Hawaii. Maybe Lonely Planet has one.